Distributed algorithms with dynamical random transitions

• Authors: Nadine Guillotin-Plantard;René Schott
• Venue: Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
• Year: 2002

We analyze a model of exhaustion of shared resources where allocation and deal-location requests are modeled by dynamical random variables as follows: Let (E, A, µ, T) be a dynamical system where (E, A, µ) is a probability space and T is a transformation defined on E. Let d ≥ 1 and f1,.....,fd be functions defined on E with values in [0, 1/d]. Let (Xi)i ≥ 1 be a sequence of independent random vectors with values in Zd. Let x ∈ E and (ej)1 ≤ j ≤ d be the unit coordinate vectors of Zd. For every i ≥ 1, the law of the random vector Xi is given by P(Xi = z) = {fj(Tix) if z = ej, 1/d - fj(Tix) if z = -ej, 0 otherwise. We write S0 = 0, Sn = ∑ sum n i=1 Xi for n ≥ 1 for the Zd-random walk generated by the family (Xi)i ≥ 1. When T is a rotation on the torus then explicit calculations are possible. This stochastic model is a (small) step towards the analysis of distributed algorithms when allocation and deallocation requests are time dependent. It subsumes the models of colliding stacks and of exhaustion of shared memory considered in the literature [14, 15, 11, 16, 17, 20]. The technique is applicable to other stochastically modeled resource allocation protocols such as option pricing in financial markets and dam management problems.